Then, take dot product \(\mathbf N\) on both sides of the equation, note \(\mathbf N\) is perpendicular to \(\sigma_u,\sigma_v, \mathbf N_u, \mathbf N_v\) and \(\mathbf N\cdot \mathbf N = 1\),
\[\Gamma_{11}^1\sigma_{uv}\cdot \mathbf N + \Gamma_{11}^2\sigma_{vv}\cdot\mathbf N + L_v = \Gamma_{12}^1\sigma_{uu}\cdot\mathbf N + \Gamma_{12}^1\sigma_{uv}\cdot\mathbf N + M_u\]
Then, note that Gauss equations represents \(\sigma_{uu},\sigma_{uv}, \sigma_{vv}\) in in terms of \(\sigma_u, \sigma_v, \mathbf N\), dot product \(\sigma_u\) so that
and re-order the equation above, we obtain the second equation in the claim.
Similarly, we can obtain 4 equations from \(\sigma_{uuv} = \sigma_{uvu}, \sigma_{uvu} = \sigma_{vvu}\) and dot product each of \(\sigma_u, \sigma_v\).
Theorem
If two surface patches has the same first and second fundamental forms, then exists a direct isometry between them.
Moveover, for \(V\subset\mathbb R^2\) open, given \(EFGLMN\) being 6 smooth functions on \(V\) and \(E>0,G>0,EG-F^2>0\) and the 6 equations (2 CM, 4 Gauss) hold. Then for any point \(p\) on \(V\), there is an open set of \(U\subset V, p\in U\) and a surface patch \(\sigma:U\rightarrow\mathbb R^3\) s.t. \(\sigma\) has the first and second fundamental forms defined by \(EFGLMN\).
In other words, given appropriate functions for first and second fundamental forms, a surface will always exist locally, and being unique up to a direct isometry.
Example: Cylinder
Let the \(E=G=1, F=0, L=-1, M=N=0\). We verify the conditions that \(E>0,G>0, EG-F^2 >0\), all coefs are zero hence \(\Gamma\)'s are all 0 and CM, Gauss are all satisifed. Therefore, a surface patch must exist.
Using Gauss equations and plug in the numbers
\[\sigma_{uu} = \Gamma_{11}^1\sigma_u + \Gamma_{11}^2\sigma_v + L\mathbf N = -\mathbf N, \sigma_{uv} = \cdots = 0, \sigma_{vv} =\cdots = 0\]
Since \(\sigma_v\)'s derivatives \(\sigma_{vu} = \sigma_{vv} = 0\), we must have that \(\sigma_v = \mathbf a\) being a constant. Since \(\sigma_{uv} = 0\), we have that
\[\sigma(u,v) = \mathbf b(u) + \mathbf a v\]
where \(\mathbf N = -\mathbf b''\) Then, consider \(\mathbf N_u,\mathbf N_v\) where \(-\mathbf N_u = a\sigma_u + b\sigma_v, -\mathbf N_v = c\sigma_u + d\sigma_v\) where
Since \(\mathbf b''' + \mathbf b' = 0\), we must also have that \(\mathbf b'' + \mathbf b = -\mathbf N + \mathbf b\) being constant vector. Take \(\mathbf b(u) = \mathbf c \sin u + \mathbf d \cos u\), and to make \(\mathbf b\) and \(\mathbf b''\) constant, take \(\mathbf c = e_1,\mathbf d = e_2\).
Finally, \(\sigma_u\times \sigma_v = \lambda \mathbf N\implies \mathbf b' \times \mathbf a = \lambda \mathbf b\implies \mathbf a = (0, 0, \lambda)\) for any \(\lambda \neq 0\), say \(\lambda =1\). Therefore, we have that \(\sigma(u,v) = (\cos u, \sin u, v)\) is a parameterization of the unit cylinder.
Example: Sphere
Let \(E = \cos^2 v, F = 0, G = 1, L =-\cos^2 v, M = 0, N = -1\).
so that \(\sigma_u = \mathbf N_u, \sigma_v = \mathbf N_v\) Therefore, \(\mathbf N = \sigma + \mathbf a\) where \(\mathbf a\) is constant and \(\|\sigma+\mathbf a\| = 1\). Since the surface patch is equivalent to the Weingarten map, hence the Gauss map, it is an isometry from the Gauss map, which is the unit sphere.
The parameterization is given by
\[\sigma(u, v) = (\cos v\cos u, \cos v\sin u, \sin v)\]
Example: Not a surface patch
\(E = 1, F = 0, G = \cos^2 u, L = \cos^2 u, M = 0, N = 1\) In first fundamental form, \(G\) is the only non-constant and \(F\) is 0. Therefore, the non-zero Christoffel symbol are
and then note that the principal curvatures are simply the roots of \((L-\kappa E)(N-\kappa G) = 0\), since \(M=F=0\), thus we have \(\kappa_1 = \frac{L}{E}, \kappa_2 = \frac{N}{G}\) so that
Take \(2\sqrt{EG}\) out of the RHS, and we can obtain the equation.
Corollary If \(E=1,F=0\), then \(K = -\frac{1}{\sqrt G}\frac{\partial^2\sqrt G}{\partial u^2}\)
Example: Sphere
Claim any plane map of any region of a sphere must distort distances.
proof. The claim is equivalent to that there is no isometry between open subset of a plane to a sphere. Known that the Gaussian curvature of plane is \(0\) everywhere, while \(K\) for sphere is positive constant. Therefore, such isometry cannot exist.
Example
Claim If \(E=e^\lambda, F=0,G=e^\lambda\), where \(\lambda\) is a smooth function of \(u,v\). Then